In Chapter 12 of The Hobbit, what motivates Bilbo to steal the cup?

This question can be answered in three different ways, all of which are true.  First, one could say that Bilbo is motivated by Thorin's words at the entrance to the cave.  Thorin spends a long time describing Bilbo as a "good companion" as a "hobbit full of courage" and "good luck" (191).  Bilbo finally admits having gotten Thorin "out of two messes already" but that the "third time pays for all" (191) and avidly begins the journey downward to Smaug's lair.  Second, one could say that it was the ring itself that finally motivated Bilbo to go farther, for Bilbo doesn't go far in the cave without slipping it on.  "Then the hobbit slipped on his ring, and . . . he crept noiselessly down, down, down into the dark" (192).  Finally, even though it isn't long before Bilbo complains "I have absolutely no use for dragon-guarded treasures," one could say that Bilbo is ultimately motivated by just that:  the lust for gold.  When Bilbo finally sees Smaug's glorious treasure, even though Bilbo had been told of it long before, "the splendor, the lust, the glory of such treasure had never yet come home to him.  His heart was filled and pierced with enchantment and with the desire of dwarves; and he gazed motionless, almost forgetting the frightful guardian, at the gold beyond price and count" (194).  It is only then that Bilbo reaches for the cup and brings it back successfully.  But is that lust Bilbo feels from his own heart, . . . or from donning the ring?  Ah, there's the rub.  For this reason, I believe the answer to your question can only be one:  the ring.